Time accurate anisotropic goal-oriented mesh adaptation for unsteady flows

We present a new algorithm for combining an anisotropic goal-oriented error estimate with the mesh adaptation fixed point method for unsteady problems. The minimization of the error on a functional provides both the density and the anisotropy (stretching) of the optimal mesh. They are expressed in terms of state and adjoint. This method is used for specifying the mesh for a time sub-interval. A global fixed point iterates the re-evaluation of meshes and states over the whole time interval until convergence of the space–time mesh. Applications to unsteady blast-wave and acoustic-wave Euler flows are presented.     

Second- and third-harmonic generations for a nondilute suspension of coated particles with radial dielectric anisotropy

We derive expressions for the effective nonlinear susceptibility tensors for both the second harmonic generation (SHG) and induced third harmonic generation (THG) of nonlinear composite materials, in which nondilute coated particles with radial dielectric anisotropy are randomly embedded in the linear host. Two types of coated particles are considered. The first is that the core possesses a second order nonlinear susceptibility and the shell is linear and radially anisotropic, while the second is that the core is linear with radial anisotropy and the shell has a second order nonlinear susceptibility. We observe greatly enhanced SHG and THG susceptibilities at several surface plasmon resonant frequencies. For the second model, due to the coating material being metallic, there exists two fundamental resonant frequencies ω_c1 and ω_c2, whose difference ω_c2-ω_c1 is strongly dependent on the interfacial parameter and the radial dielectric anisotropy. Furthermore, in both systems, the adjustment of the dielectric anisotropy results in larger enhancement of both SHG and induced THG susceptibilities at surface plasmon resonant frequencies than the corresponding isotropic systems. Therefore, both the core-shell structure and the dielectric anisotropy play important roles in determining the nonlinear enhancement and the surface resonant frequencies.     

A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates

A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on hierarchical a posteriori error estimates. A global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution. Instead of solving the global error problem exactly, which is costly in general, we solve it iteratively using the symmetric Gauß–Seidel method. Numerical results show that a few GS iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation. The new method is compared with several strategies using local error estimators or recovered Hessians. Numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients.     

Mesh locking effects in the finite volume solution of 2-D anisotropic diffusion equations

Strongly anisotropic diffusion equations require special techniques to overcome or reduce the mesh locking phenomenon. We present a finite volume scheme that tries to approximate with the best possible accuracy the quantities that are of importance in discretizing anisotropic fluxes. In particular, we discuss the crucial role of accurate evaluations of the tangential components of the gradient acting tangentially to the control volume boundaries, that are called into play by anisotropic diffusion tensors. To obtain the sought characteristics from the proposed finite volume method, we employ a second-order accurate reconstruction scheme which is used to evaluate both normal and tangential cell-interface gradients. The experimental results on a number of different meshes show that the scheme maintains optimal convergence rates in both L² and H¹ norms except for the benchmark test considering full Neumann boundary conditions on non-uniform grids. In such a case, a severe locking effect is experienced and documented. However, within the range of practical values of the anisotropy ratio, the scheme is robust and efficient. We postulate and verify experimentally the existence of a quadratic relationship between the anisotropy ratio and the mesh size parameter that guarantees optimal and sub-optimal convergence rates.     

流动数值模拟中一种并行自适应有限元算法

给出了一种流动数值模拟中的基于误差估算的并行网格自适应有限元算法.首先,以初网格上获得的当地事后误差估算值为权,应用递归谱对剖分方法划分初网格,使各子域上总体误差近似相等,以解决负载平衡问题.然后以误差值为判据对各子域内网格进行独立的自适应处理.最后应用基于粘接元的区域分裂法在非匹配的网格上求解N-S方程.区域分裂情形下N-S方程有限元解的误差估算则是广义Stokes问题误差估算方法的推广.为验证方法的可靠性,给出了不可压流经典算例的数值结果.     

Design of MMICs employing whispering-gallery mode dielectric resonators at microwave and millimeter wave frequencies

A simple theoretical model for predicting the resonant frequencies of the modes of the whispering — gallery (WG) mode dielectric resonators MICs (Microwave Integrated Circuits) at microwave and millimetre wave frequencies is presented. The unsymmetric field distribution for WGE mode in the axial direction is taken into account. The theoretical results are found to be in good agreement with the practical results taken at microwaves C-X bands. In addition, the variation of resonant modes against dielectric resonator's dimensions and dielectric constant is modelled. The effect of the width of dielectric substrate on resonant modes is also illustrated.     

Simulating finger phenomena in porous media with a moving finite element method

The non-equilibrium Richards equation is solved using a moving finite element method in this paper. The governing equation is discretized spatially with a standard finite element method, and temporally with second-order Runge–Kutta schemes. A strategy of the mesh movement is based on the work by Li et al. [R.Li, T.Tang, P.W. Zhang, A moving mesh finite element algorithm for singular problems in two and three space dimensions, Journal of Computational Physics, 177 (2002) 365–393]. A Beckett and Mackenzie type monitor function is adopted. To obtain high quality meshes around the wetting front, a smoothing method which is based on the diffusive mechanism is used. With the moving mesh technique, high mesh quality and high numerical accuracy are obtained successfully. The numerical convergence and the advantage of the algorithm are demonstrated by a series of numerical experiments.     

Asymptotic analysis of nonlinear nonaxisymmetric waves on the surface of a neutral dielectric jet in a longitudinal electrostatic field

The problem of calculation of finite-amplitude waves on the cylindrical surface of an ideal incompressible dielectric liquid jet in a uniform electrostatic field collinear to the unperturbed jet axis is solved using a second-order asymptotic analytic procedure in ratio of the wave amplitude to the jet radius. Nonlinear corrections to the jet profile, velocity field potential, and electrostatic potentials inside and outside the jet are of resonant nature. The degenerate resonant interaction between the wave determining the initial strain and the waves excited due to nonlinearity of the hydrodynamic equations can take place for waves with different symmetries (different azimuth numbers).     

一维反铁磁光子晶体光学双稳态效应研究

基于传递矩阵方法和光局域化原理, 本文研究了一维反铁磁光子晶体共振频率附近光学双稳态效应随电磁波入射角、 外磁场强度及电介质层数的变化.研究发现, 当外磁场为1.0 kG(1 G=10^-4 T)、电磁波小角度入射时, 反铁磁材料高低共振频率附近均可探测到光学双稳态效应; 当电磁波大角度入射时, 仅在高共振频率附近探测到光学双稳态效应. 然而, 当外磁场强度增加到2.0 kG时, 由于反铁磁材料的高低共振频率向两侧移动, 低共振频率附近缺失的光学双稳态频率窗口又被有效激发. 此外, 电磁波小角度入射时, 电介质层数在低共振频率附近对双稳态效应影响较明显.     

Finite difference scheme for planar waveguides with arbitrary index profiles and its implementation for anisotropic waveguides with a diagonal permitivity tensor

A finite difference scheme with a uniform mesh for planar waveguides with arbitrary refractive index profiles that takes full account of any smooth index variation and index discontinuity is derived for TE and TM-polarized waves. Discretizations that lead to a second-order error in the effective indices are given for TE and TM polarizations. At the computational boundaries, transparent boundary conditions are used. The scheme was implemented for anisotropic waveguides with a diagonal permitivity tensor and examined by using samples with various refractive index profiles, ranging from simple step- and graded-index up to complicated refractive index profile structures composed of either isotropic or anisotropic materials. For simple cases where the results of other methods are available in the literature, the proposed scheme shows very good agreement.     

High order numerical approximation of minimal surfaces

We present an algorithm for finding high order numerical approximations of minimal surfaces with a fixed boundary. The algorithm employs parametrization by high order polynomials and a linearization of the weak formulation of the Laplace–Beltrami operator to arrive at an iterative procedure to evolve from a given initial surface to the final minimal surface. For the steady state solution we measure the approximation error in a few cases where the exact solution is known. In the framework of parametric interpolation, the choice of interpolation points (mesh nodes) is directly affecting the approximation error, and we discuss how to best update the mesh on the evolutionary surface such that the parametrization remains smooth. In our test cases we may achieve exponential convergence in the approximation of the minimal surface as the polynomial degree increases, but the rate of convergence greatly differs with different choices of mesh update algorithms. The present work is also of relevance to high order numerical approximation of fluid flow problems involving free surfaces.     

Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing

Metric tensors play a key role to control the generation of unstructured anisotropic meshes. In practice, the most well established error analysis enables to calculate a metric tensor on an element basis. In this paper, we propose to build a metric field directly at the nodes of the mesh for a direct use in the meshing tools. First, the unit mesh metric is defined and well justified on a node basis, by using the statistical concept of length distribution tensors. Then, the interpolation error analysis is performed on the projected approximate scalar field along the edges. The error estimate is established on each edge whatever the dimension is. It enables to calculate a stretching factor providing a new edge length distribution, its associated tensor and the corresponding metric. The optimal stretching factor field is obtained by solving an optimization problem under the constraint of a fixed number of edges in the mesh. Several examples of interpolation error are proposed as well as preliminary results of anisotropic adaptation for interface and free surface problem using a level set method.     

基于P范数的核最小对数绝对差自适应滤波算法

为了进一步提高在a稳定分布噪声背景下非线性自适应滤波算法的收敛速度,本文提出了一种新的基于p范数的核最小对数绝对差自适应滤波算法(kernel least logarithm absolute difference algorithm based on p-norm, P-KLLAD).该算法结合核最小对数绝对差算法和p范数,一方面利用最小对数绝对差准则保证了算法在a稳定分布噪声环境下良好的鲁棒性,另一方面在误差的绝对值上添加p范数,通过p范数和一个正常数a来控制算法的陡峭程度,从而提高该算法的收敛速度.在非线性系统辨识和Mackey-Glass混沌时间序列预测的仿真结果表明,本文算法在保证鲁棒性能的同时提高了收敛速度,并且在收敛速度和鲁棒性方面优于核最小均方误差算法、核分式低次幂算法、核最小对数绝对差算法和核最小平均p范数算法.     

Radiation properties of microstrip patch antenna covered with an anisotropic dielectric layer and a plasma sheath

By using SO-FDTD method, radiation properties of microstrip patch antenna covered with an anisotropic dielectric layer and a plasma layer are investigated. Simulation results show that the resonant frequency of antenna covered with a plasma layer varies with different plasma parameters. It has been indicated that the frequency offset of antenna is as high as 120 MHz when plasma frequency changes from 1 GHz to 8 GHz. The effect on antenna covered with an anisotropic dielectric layer is analyzed, while altering the dielectric constant or thickness of dielectric layer. Finally, the peak gain of antenna in complex plasma environment is presented in detail.     

基于数据重建的三角网格模型简化优化方法

由CT切片数据重建得到的三角网格模型常存在数据量大、狭长三角形多等问题,针对这些问题,研究了一种保持特征的高质量三角网格模型的简化优化方法。该方法分为网格简化和网格优化两个阶段。首先,采用基于曲面变化的二次误差度量计算边折叠代价,并按代价值的大小进行迭代的折叠简化,可较好地保持模型表面的特征;其次,通过二阶加权伞算子对简化模型中局部存在的狭长三角形进行优化处理,改善三角网格模型的质量。实验结果表明,该方法能够较好地保持特征区域的细节信息,并可靠地生成高质量、低几何误差的简化模型。     

微波脉冲与带介质孔缝矩形腔体耦合的数值模拟研究

利用时域有限差分(FDTD)方法对微波脉冲与带介质孔缝矩形腔体的耦合过程进行了数值模拟研究. 如果孔缝填有介质, 则预期对微波耦合进入腔体的物理过程有重要影响, 研究了微波与带介质孔缝矩形腔体耦合的过程中影响介质孔缝耦合共振峰和共振频率点的因素, 包括孔缝长度、宽度和介质相对介电常数等物理量的影响. 通过大量不同孔缝尺寸的模拟研究, 发现孔缝的介质对孔缝耦合共振频率有明显影响, 我们对微波与带介质孔缝耦合发生共振的公式进行了拟合, 最后得出了微波脉冲与带介质孔缝矩形腔体耦合的共振条件.     

Two-Level Newton Iteration Methods for Navier-Stokes Type Variational Inequality Problem

This paper deals with the two-level Newton iteration method based on the pressure projection stabilized finite element approximation to solve the numerical solution of the Navier-Stokes type variational inequality problem. We solve a small Navier-Stokes problem on the coarse mesh with mesh size $H$ and solve a large linearized Navier-Stokes problem on the fine mesh with mesh size $h$. The error estimates derived show that if we choose $h=\mathcal{O}(|\log h|^{1/2}H^3)$, then the two-level method we provide has the same $H^1$ and $L^2$ convergence orders of the velocity and the pressure as the one-level stabilized method. However, the $L^2$ convergence order of the velocity is not consistent with that of one-level stabilized method. Finally, we give the numerical results to support the theoretical analysis.     

Normal modes and quality factors of shielded composite dielectric spherical resonators

Electromagnetic field analysis of shielded composite dielectric spherical resonator in which the dielectric sphere is composed of two concentric spheres with different dielectric materials has been made. Characteristic equations for the TE_nmℓ and TM_nmℓ modes have been derived. From these characteristic equations, the resonant frequencies and quality factors have been calculated using numerical method. Computations of the resonant frequencies and quality factors have been made for resonators with parameters suitable for the optical and microware regions. In this paper we have presented the analysis and the results obtained from the numerical computations for shielded composite dielectric spherical resonator in which the dielectric sphere is composed of two concentric spheres with different dielectric materials. Starting from the Maxwell’s equations for such a resonators have been derived and resonant frequencies and quality factor’s have been calculated for the TE_nmℓ and TM_nmℓ modes using numerical method.     

Dispersion characteristics of asymmetric coupled anisotropic dielectric waveguides using FDFD

The formulation is developed in the frequency domain and the finite difference method is used for the numerical solution of the scalar wave equation, written in terms of the transverse components of the magnetic field. As a result a conventional eigenvalue problem is obtained without the presence of spurious modes due to the implicit inclusion of the divergence of the magnetic field equal to zero. The formulation is developed to include biaxial anisotropic dielectrics with an index profile varying arbitrarily in the cross section of the waveguide under analysis. This formulation is then applied to the analysis of the influence on the dispersion characteristics of the dimensions of asymmetric coupled rectangular uniaxial anisotropic dielectric waveguides. As expected, the reduction of the height or the width of one of the rectangular dielectric waveguides causes the dispersion curves to move towards higher frequencies.     

双层粘接板界面的超声非线性谐振特性分析

采用声-力-电类比建立粘接界面非线性力学行为的等效非线性振荡电路,以求解双层粘接板的超声非线性谐振频率。理论上导出非线性谐振频率方程,确定双层粘接板非线性谐振频率与激励幅度、三阶弹性劲度系数的解析关系。双层粘接铝板的超声实验发现:在不良粘接情形下,超声谐振频率发生偏移,其值大于粘接完好区,且激发了较强的三次谐波,但二次谐波幅度变化不大。实验结果表明三次谐波幅度上升,超声谐振频率也显著增大,与理论导出非线性谐振频率变化规律相吻合,且三次谐波与基波、二次谐波的比值反映了非线性谐振频率变化趋势,证实粘接层三阶劲度系数是产生非线性共振频率偏移的主要因素。